Contact interactions on graph superlattices

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or delta-type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the delta' interaction and its symmetrized version for a particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig-Penney analogues: the delta' lattices have bands whose widths are asymptotically bounded and do not approach zero, while tile delta lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the delta coupling constant is small enough, the delta lattice has no gaps above the threshold of the spectrum. On the other hand, infinitely many gaps emerge above a critical value of the coupling constant; for almost all ratios this value is zero.